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I'm not a mathematician, but a part time artist, and I enjoy drawing patterns. I came up with this drawing a couple years ago:

The full design

Someone may have come up with this before me, I wouldn't be surprised. I don't know if it's a fractal, but as it is drawn it branches out in a way which reminds me of fractals. I'd like to know if anyone has any explanation for this type of pattern or knows of similar patterns that I could compare it to. Thanks.

The rules for the design are as follows:

  • A drawn arch ends as it intersects any part of another arch.
  • As an arch [A] completes one quarter of a circle's circumference, (if it is not at an intersection) it branches out into two more arches, where one branch [B] continues to complete another quarter of the same circle's circumference and the other branch [C] moves out in a reflected direction described by an imaginary axis extending from the end of arch [A]

The first step in drawing the design is to select a center within four quadrants, having a horizontal and vertical axis. Four initial archs branch out from the center in a circular curve, connecting (0,0) to (1,1) in the first quadrant, (0,0) to (-1,1) in the second quadrant, (0,0) to (-1,-1) in the third quadrant, and (1,-1) in the fourth quadrant. It's easier to understand when shown through a visual representation:

The first three steps are shown here

As it is drawn it seems to develop squares at the corners or existing squares.

Jacob Manaker
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sam matney
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    From a first glance, this seems to be related to the toothpick fractal: https://en.wikipedia.org/wiki/Toothpick_sequence – person May 02 '22 at 20:22
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    It reminds me of this video from Numberphile. – Arthur May 02 '22 at 20:25
  • Thank you both so much, I was really curious. – sam matney May 02 '22 at 20:52
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    It's almost identical to the gull wing toothpick fractal from the numberphile video, except that I drew it as four archs emerging from the center, whereas omar Pol (the creater of the fractal) started with the upper half of my first step and continued the design from there. As a result, his fractal has identical designs along the horizontal and vertical axes. In contrast, the horizontal axes of my fractal produces a constant unchanging design, and only the designs along the vertical axes develop. Aside from that, I believe they are basically identical. – sam matney May 03 '22 at 08:54

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